Minimum rank

The M nonzero eigenvalues of both matrices Qa and Qb coincide (M being the minimum rank between the two Q matrices) and twice the sum of the square root of these eigenvalues X is the interset common variance Vab ... 

Remarkable differences can be noted. Thus, in the case of the three sites 1, 17, and 91 rather sharp maxima are developed, indicating that they can safely be assigned to a rank near the maximum of their probability plot. However, the sites differ in their individual ranking position. Thus, site 91 takes a lower rank site 1 a medium rank and site 17 a rather high rank. Therefore a mutual ranking sequence of the sites 1, 17, and 91, i.e., 91 < 1 < 17, can be given since the minimum rank of the one site apparently does not overlap significantly with the maximum rank of a lower positioned site. 

Therefore, the focus shifts from indicator to landscape unit (watershed in this case). Statistics are computed on the set of rank values for each landscape unit (watershed). Statistics needed for the present operation are minimum rank, maximum rank, and median rank. These become three derived observational variables for each landscape unit (watershed). Given these preparatory computations, the operations for rank range sequencing are as follows ... 

The dielectric constant is also affected by stmctural changes on strong heating. Also the value is very rank dependent, exhibiting a minimum at about 88 wt % C and rising rapidly for carbon contents over 90 wt % (4,6,45). Polar functional groups are primarily responsible for the dielectric of lower ranks. For higher ranks the dielectric constant arises from the increase in electrical conductivity. Information on the freedom of motion of the different water molecules in the particles can be obtained from dielectric constant studies (45). 

Each maximum, minimum or saddle point occurs at a so-called critical point Tc, where the gradient vanishes. The nature of the critical point is determined by the eigenvalues of the Hessian. All the eigenvalues are real at the critical point, but some of them may be zero. The rank co of the critical point is defined to be the number of non-zero eigenvalues. The signature o is the sum of the signs of the eigenvalues, and critical points are discussed in terms of the pair of numbers (w, o). 

Let s compare these plots of the REV s to the plot in Figure 52. Notice that these REV s do not exhibit ideal behavior. Ideally, as rank increases, the REV s would drop to some minimum value and then remain at that level. These REV s begin to tail back up. This sort of non-ideal behavior is not uncommon when working with actual data. Unfortunately, it can complicate matters when we use the 2-way F-test to see which REV s represent basis vectors and which ones represent noise vectors. 

It has been shown that the p columns of an nxp matrix X generate a pattern of p points in 5" which we call PP. The dimension of this pattern is called rank and is indicated by liPP). It is equal to the number of linearly independent vectors from which all p columns of X can be constructed as linear combinations. Hence, the rank of PP can be at most equal to p. Geometrically, the rank of P can be seen as the minimum number of dimensions that is required to represent the p points in the pattern together with the origin of space. Linear dependences among the p columns of X will cause coplanarity of some of the p vectors and hence reduce the minimum number of dimensions. 

Example 11.2 Using the Underwood Equations, determine the best distillation sequence, in terms of overall vapor load, to separate the mixture of alkanes in Table 11.2 into relatively pure products. The recoveries are to be assumed to be 100%. Assume the ratio of actual to minimum reflux ratio to be 1.1 and all columns are fed with a saturated liquid. Neglect pressure drop across each column. Relative volatilities can be calculated from the Peng-Robinson Equation of State with interaction parameters assumed to be zero (see Chapter 4). Determine the rank order of the distillation sequences on the basis of total vapor load for ... 

In this case study, 11 different extraction methods (Table 1) were applied to a 1000 m line of 15 B/C-horizon soil samples (<0.25 mm) across the Talbot VMS Cu-Zn deposit in northern Manitoba, Canada (Fig. 1), followed by ICP-MS analysis. Student s t test and minimum t probability statistics provide a tool to rank the exploration performance of each method (Stanley 2003 Stanley Noble 2008). By... 

Your minimum acceptable rate of return (MARR) is 18%, the project life is 10 years, and no alternatives have a salvage value. The following mutually exclusive alternatives have been proposed. Rank them, and recommend the best alternative. 

You have four choices of equipment (as shown in the following table) to solve a pollution control problem. The choices are mutually exclusive and you must pick one. Assuming a useful life of 10 years for each design, no market value, and a pretax minimum acceptable rate of return (MARR) of 15% per year, rank them and recommend a choice. 

The nature of a critical point is denoted as (rank, signature). For example, a minimum in p is designated by (3, + 3) and a maximum by (3,-3). The two remaining types of critical points, (3,-1) and (3, +1) are saddle points, called bond critical point and ring critical point, respectively. 

The question of exactly how many coefficients are required to describe the conserved-scalar field does not end with (5.68), and is a significant one since it will determine the minimum number of transport equations that must be solved to describe the flow completely. From the form of (5.68), it can be seen that tpc is alinear combination of the columns of d 4JY43 Thus, only a linearly independent sub-set of the columns is required to describe ipc completely. The exact number is thus equal to N m = rank(i0)) < mini A, Nm). 

The technique of H NMR thermal analysis (25) yields data In the form of M2J pyrograms (typical examples are shown In Figures 1 and 2). The fusibility of a coal can be ranked by the minimum value of M2J (M2jmln) attained during the experiment. This parameter measures the Instantaneous maximum extent of fusion of the sample on a molecular level and not the total fraction of material that passes through a fused state during heating. 

The M2.J, pyrograms for the pure Inertlnltes representative of the two lower rank coal subsets (Figures 3 and 4) Indicate little molecular mobility In the temperature region of coal thermoplasticity (l.e. 700 K). However, the well defined minimum... 

---